SOME FURTHER PROPERTIES OF SOFT SUBGROUPS

A subgroup H of a p-group G is n-uniserial if for each i = 1,...,n, th ere is a unique subgroup K-i such that H less than or equal to K-i and \K-i:H\ = p(i). In case the subgroups of G containing H form a chain we say that H is uniserially embedded in G. We prove that if p is odd and that K is a 2-uniserial subgroup of order p in the p-group G. Then K is uniserially embedded in G. We also show that for p > 3, if K is a 2-uniserial cyclic subgroup of the p-group G, then K is uniserially embedded in G. We prove the following two theorems: (1) Let A be a sof t subgroup of index greater than p. Let N-1 = N-G(A) and R = G'Z(N-1). Then the A-invariant subgroups of R containing Z(N-1) form a chain. ( 2) Suppose that the p-group G has a uniserially embedded subgroup P of order p. Then either G has a cyclic subgroup of index p or is of maxi mal class (coclass 1).